JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 3, July 1989

AFFINE HECKE ALGEBRAS AND THEIR GRADED VERSION

GEORGE LUSZTIG Dedicated to Sir Michael Atiyah on his sixtieth birthday

INTRODUCTION

0.1. Let Hvo be an affine Hecke algebra with parameter Vo E C* assumed to be of infinite order. (The basis elements ~ E Hvo corresponding to simple reflections s satisfy (~+ 1)(Ts - v~C(s») = 0, where c(s) EN depend on s and are subject only to c(s) = c(s') whenever s, Sf are conjugate in the affine Weyl group.) Such Hecke algebras appear naturally in the representation theory of semisimple p-adic groups, and understanding their representation theory is a question of considerable interest. Consider the "special case" where c(s) is independent of s and the coroots generate a direct summand. In this "special case," the question above has been studied in [I] and a classification of the simple modules was obtained. The approach of [1] was based on equivariant K-theory. This approach can be attempted in the general case (some indications are given in [5, 0.3]), but there appear to be some serious difficulties in carrying it out. 0.2. On the other hand, in [5] we introduced some algebras H,o' depending on a parameter roE C , which are graded analogues of Hvo . The graded algebras

H '0 are in many respects simpler than Hvo' and in [5] the representation theory of H,o is studied using equivariant homology. Moreover, we can make the

machinery of intersection cohomology work for us in the study of H '0 ' while in the K-theory context of Hvo it is not clear how to do this. In particular, the

difficulties mentioned above disappear when Hvo is replaced by H '0 • For this reason, it seemed desirable to try to connect the representation the- ories of Hvo and H,o' In this paper, we shall prove that the classification of simple Hvo -modules can be reduced to the same problem, where Hvo is replaced essentially by H,o' This makes it possible to study the representation theory of Hvo without K-theory but with the aid of H,o' (This is analogous to studying the representations of a Lie group using the theory of Lie algebras.) It should

Received by the editors March 14, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A89, 16A03.

© 1989 American Mathematical Society 0894-0347/89 $ 1.00 + $.25 per page

599

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be pointed out that this approach fails when Vo is a root of 1, other than 1. This is a very interesting case which has been studied very little. 0.3. In an unpublished work, Bernstein (partly in collaboration with Zelevin- skii) described a large commutative subalgebra of Hvo and the center of Hvo (in the "special case" above). We extend these results to the general case. A number of new difficulties arise, most of them caused by the simple coroots which are divisible by 2. This extension is done in §3, based on the preparation of §§1 and 2. In §4, we introduce a filtration of an affine Hecke algebra and show that the corresponding graded algebra is the one introduced in [5]. §§5-7 are again preparatory. In §8, we show that the completion of an affine Hecke algebra with respect to a maximal ideal of the center is isomorphic to the ring of n x n matrices with entries in the completion of a (usually smaller) affine Hecke alge- bra. In §9, we define a natural homomorphism from an affine Hecke algebra to a suitable completion of its graded version, which becomes an isomorphism when the first algebra is completed. This homomorphism is of the same nature as the Chern character from K-theory to homology. In §1O, we combine the results of §§8 and 9 to get our main result (see 10.9), comparing the representation theory of an affine Hecke algebra with that of a graded one. 0.4. Most of the work on this paper was done during the Fall of 1988 when I enjoyed the hospitality and support of the Institute for Advanced Study, Prince- ton. I acknowledge partial support of N.S.F. grant DMS-8610730 (at lAS) and DMS-8702842.

CONTENTS

1. Root systems and affine Weyl groups 2. The braid group 3. The Hecke algebra 4. The graded Hecke algebra 5. The elements tw and Tw 6. Some braid group relations 7. Completions 8. First reduction theorem 9. Second reduction theorem 10. On simple H -modules References

1. ROOT SYSTEMS AND AFFINE WEYL GROUPS

1.1. A root system (X,Y,R,R,n) consists of (a) two free abelian groups of finite rank X, Y with a given perfect pairing (,}:XxY--+Z,

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(b) two finite subsets ReX, R c Y with a given bijection R +-+ R denoted 0: +-+ a , and (c) a subset II cR. These data are subject to requirements (d)-(f) below. (d) (0: ,a} =2 for all o:ER. (e) For any 0: E R, the reflection sa: X -t X, X t-+ X - (x, a}o: (resp. Sa: Y -t Y, y -t y - (o:,y}a) leaves stable R (resp. R). (f) Any 0: E R can be written uniquely as 0: = E pEn na,p • p, where na ,p E Z are all ~ 0 or all :::; O. (Accordingly, we say that 0: E R+ or 0: E R- .) Throughout this paper we will fix a root system (X, Y ,R, R, ll) . The subgroup of GL(X) generated by all sa (0: E R) may be identified with

the subgroup of GL(Y) generated by all sa (0: E R) by w -t tw-I . This is the Weyl group Wa of the root system. It is a finite Coxeter group on generators {salo: E ll}.

1.2. Consider the partial order:::; on R defined by a l :::; a 2 <:} a 2 - a l is a linear combination with integer ~ 0 coefficients of elements of {alo: E ll} . Let llrn be the set of all PER such that /J is a minimal element for :::;. 1.3. We shall assume throughout the paper that (X, Y ,R, R, ll) is reduced, i.e., 0: E R => 20: (j. R . 1.4. Let W be the semidirect product Wa' X. Its elements are wax (w E f ,-1, W, X E X) with multiplication given by wax. w' aX = ww' aW (X)+X; a is a fixed symbol. W acts on Y x Z by wax: (y, k) -t (w(y) , k - (x, y}). Let ~ ~+ ~ R = R U R- c Y x Z be defined by R+ = {(a,k)lo: E R, k > O} U {(a , 0)10: E R+}, R- = {(a,k)lo:ER, k

Let fi = {(a ,0)10: E ll} U {(a, 1)10: E llrn} c R+, S = {salo: E ll} U {saaalo: E llrn} C W. We have an obvious bijection fi +-+ S (A +-+ SA)' An element of S maps the corresponding element of fi to its negative and it maps the complement of this

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element in R+ into itself. It follows that for z E W, A E fi, we have (b) I(SAZ) = { I(z) + 1, if z-I(A) E R+, I(z) - 1, if z-I(A) E R- . It is clear that (c) I(w) = I(w-I) (w E W). We deduce that for x EX, a E TI, we have (dl) {x ,a} > 0 => l(soaX) = I(ax) + 1, l(aXso) = I(ax) - 1. (d2) {x ,a} < 0 => l(soaX) = I(ax) - 1, I(axso) = I(ax) + 1. (d3) {x ,a} = 0 => I(soax) = I(x) + 1, l(aXso) = I(x) + 1. (e) Let

Xdom = {x E XI{x ,a} ~ 0, Va E TI} = {x E XI/(soax) = I(ax) + 1, Va E TI}. We have, using (a) and (e), (f) WE Wa, x E Xdom => I(ax) = EoER+{X ,a}, I(wax) = I(w) + I(ax). It follows that I x X' X X' (g) X,X EXdom=>/(a ·a )=/(a )+/(a ). 1.5. Let Q be the subgroup of X generated by R. The subgroup WoQ of W is a Coxeter group with S as the set of simple reflections, the length function being the restriction of I. This subgroup is normal in Wand admits a complement n = {w E Wlw(ll) = fi} = {w E WI/(w) = O}. It is known that n is an abelian group, isomorphic to X / Q . 1.6. The notion of direct sum of root systems is defined in an obvious way. We say that (X, Y, R, R, TI) is primitive if either (a) or (b) below holds. (a) a ¢ 2Y for all a E R. (b) There is a unique a E TI such that a E 2Y; moreover, {w(a)lw E Wo} generate X.

Lemma 1.7. In general, (X, Y,R,R,TI) is (uniquely) a direct sum a/primitive root systems. Proof. We may assume that we can find a E TI such that a E 2Y. Let X' be the subgroup of X generated by {w(a)lw E Wa}, Y' the subgroup of Y generated by Hw(a)lw E Wa}, X" = {x E XI{x ,y'} = 0, Vy' E Y'}, y" = {y E YI{x' ,y} = 0, Vx' EX'}.

It is easy to see that { , } defines by restriction a perfect pairing X' x Y' --+ Z . Hence, we have X = X' $ X", Y = Y' $ y" ,and { , } also defines a perfect pairing X" x y" --+ Z. Let R' = R n X', R' = R nY', TI' = TI n X', R" = Rnx" , R" = Rn y" , TI" = TInX" . It is clear that (X, Y ,R, R, TI) is a

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duect. sum 0 f t h e root systems ('X, Y I ,R I ,R• I , n/) and ("X, Y " ,R"." ,R, n") , the first of which is primitive. We can repeat this procedure if necessary and the lemma follows.

2. THE BRAID GROUP

2.1. Let B be the group with generators Tw (w E W) and relations (a) TWTW1 = Tww 1 whenever /(w) + /(w' ) = /(ww' ). We say that B is the braid group of W.

Lemma 2.2. Let x EX, 0 E n be such that (x, 0) = O. Let s = sa' Then Tax r: = TsTax in B. Proof. Using 2.1(a) and 1.4(d3), we have Taxr: = Taxs = r:ax = TsTax ,

Lemma 2.3. Let x E Xdom ' 0 E n be such that (x, 0) = p > 1. Let s = sa' W = sax sax. Then (a) /(ax s) = /(ax ) - 1 . (b) /(axsax) = 2/(ax) - 2p + 1. pa (c) /(w) = 21(aX) - 2p, w = a2x- , and 2x - po E Xdom ' (d) If p = 1, then r:-lTaxr:-lTax = Tw in B. Proof. Let l(x) = A.. Now (a) follows from 1.4(dl). We have saX sax = as(x)+x = a2x- pa , (2x - po, '/J) = 2(x ,'/J) - p(o, '/J) (P E n). If P ::/= 0, then (0, '/J) ~ 0 so (2x - po ,'/J) ~ O. If P = 0, then (2x - po ,'/J) = 2p - 2p = 0 . Hence, 2x - 0 E Xdom ' By 1.4(f) we have l(saxsax) = L (s(x)+x,'/J) PER+

PER+ PER+

PER+ PER+ = 2 L (x ,'/J) - 2(x ,0) PER+ = 21(ax) - 2p = 2A. - 2p, hence (c). Now (b) follows from (c) using 1.4(dl). From (a)-(c) we have (for p = 1)

hence (d). 2.4. We shall regard 8 as the set of vertices ofthe Coxeter graph of (UQQ, 8) in the usual way. For 0 En, let 8(0) C 8 be the connected component of

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s = Sa in this graph. Assume that a E 2Y. Then S(a) is a Coxeter graph of affine type C, (r ~ 1). Hence, it has a unique nontrivial automorphism -. Let S = sa E S(a). We can now state Lemma 2.S. In the setup of 2.4, there is a unique element x E Q such that (x, a) = 2, (x, /J) = 0 for all P E TI - {a}. Moreover, there exist elements w I ,w " E »OQ c W sue h t hat 2 Taxsax = Tw'w'" Ta2X - o = Tw,TsTw'" TaxsTax = Tw,Ts Tw" (in B). Proof. We may assume that X = Q and S = S(a) (see 2.4). Let YI be the subgroup (of index 2) of Y generated by R. Let XI = Hom(YI ,Z). Then X is naturally a subgroup (of index 2) of XI . Clearly, (XI' YI ,R, R, TI) is a root system. We denote WI = »0 . XI with I: ~ -+ N as in 1.4(a) extending I: W -+ N and let B I be the corresponding braid group. Then W c ~, Be BI in a natural way; moreover, if n denotes the set of elements of length o of WI' then n = {I , w} is a group of order 2 which is a complement of W in ~ and {I, Tw} is a complement of B in B I • Now conjugation by w is the nontrivial automorphism of S (as a graph). Hence, WSW-I = S (see 2.4). Let XI E XI be defined by (xl' a) = 1, (xl' /J) = 0 for pEn - {a}. We have XI ¢ X. Indeed, if we had XI EX, then from a E 2Y it would follow that (XI' a) E 2Z, contradicting (XI' a) = 1. Set X = 2xI . Then X EX, since X has index 2 in XI . Moreover, (X, a) = 2, (x, p) = 0 for pEn - {a} . Let w' - aXI sw E WI' w" E wa3xI -a E ~ . Since XI E XI - X, we have aXI E wW, a3xI E wW . Hence, w', w" E W. We set II = l(xl ). Applying Lemma 2.3 to XI instead of x, we see that (a) I(w' ) = l(axI s) = II - 1 . (b) I(w") = l(axI . a2XI -a) = l(axI ) + l(a2xl -a) (see 1.4(g)) = II + 2/1 - 2 = 3/1 - 2 . (c) I(ax ) = l(axI . aXI ) = 2/(axI ) = 2/1 (see 1.4(g)). (d) I(axls) = I(ax ) - 1 = 2/1 - 1 (see 1.4(dl)). (e) I(axls) = l(axI ) - 1 = II - 1 (see 1.4(dl)). (f) l(a2x-a) = l(a2XI .a2XI -a) = l(a2XI)+/(a2XI-a) = 2/1 +2/1 -2 = 411-2. Applying Lemma 2.3(b) to x, we see that (g) I(axsax ) = 4/1 - 1 - 3. Next we note '" XI 3xI -a XI S(XI) 2xI -a W W = a sa = a a sa

2xI 2xI X X = a sa = a sa By (a), (b), and (g) we have I(w' ) + I(w") = I(axsax) = 4/1 - 3. Hence, Tw' Tw" = Taxsax .

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We have

By (a), (b), and (f) we have I(w' ) + I(s) + I(w") = l(a2X -a) = 4/1 - 2. We have

TaxsTax = TaxI TaxIs TaxI TaxI (see (d), (e)) -I = TaxI TaxI Ts TaxI TaxI (see 1.4(dl)) = TaxI ~Tsaxlsaxl (see Lemma 2.3(d)) = TaxI Ts Tsaxlsaxl aXI (see 1.4(g)) = TaXlsTsTsTa3XI-a (see 1.4(dl)) = TWITwTsTsTwTwl! 2 = TWITs Twl!' The lemma is proved.

2.6. We define, for any x EX, an element T x E B as follows. We write x = X I -X2 with XI' x2 E Xdom ' and we set Tx = TaxI Ta-:21 . This is independent of the choice of XI ' x 2 since Taxi TaxI! = TaxI! Taxi = Taxl+xl! for x' , x" E X dom (see 1.4(g)).

Lemma 2.7. (a) If X E Xdom ' then Tx = Tax'

(b) We have T x ' Txl! = TXI+XI! for x' ,x" EX. (c) If x EX, a E n satisfy (x, a) = 0 and Sa = s, then ~ T x = T x Ts . (d) If x EX, a E n satisfy (x ,a) = 1 and sa = s, then Tx = ~Ts(X)Ts' (e) If x E X, a E n satisfy (x,a) = 2, a E 2Y, and sa = s, then I hh - I - I th ere exist e1 ements y, y E B sue t at ~Ts(x) = yy , T x _ a = yTsY , TxTs-I = YTs2y'; (s as in 2.4). Proof. (a) and (b) follow immediately from the definition. In (c), we can write x = XI - x 2 ' where XI' x 2 E Xdom satisfy (XI' a) = (X2 ' a) = 0, and it remains to observe that TaXI ' TaX2 commute with Ts by Lemma 2.2. In (d), we can write x = XI - x2 ' where XI' x2 E Xdom satisfy (XI' a) = 1 , (X2 ,a) = O. We have -1- -1- -I -I -I -I Ts T x Ts T x = Ts TaXI TaX2 Ts TaXI TaX2 -I -I -2. c = Ts TaXI Ts TaXI TaX2 (usmg Lemma 2.2 lor X 2) -I = Tsaxlsaxl Ta2X2 (using Lemma 2.3(d)) -I = Ta2XI -a Ta2X2

= T2xl-a-2x2 = T 2x- a (see Lemma 2.3(c)) = TS(X) • Tx (see (b)). --I We now multiply on the right by Tx and (d) follows.

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We fix a E II with a E 2Y . Assume that (e) holds for some x E X with (x, a) = 2 . Now let x' E X be such that (x', a) = O. Then (e) holds for x + x' instead of x, with Tx'y, y' instead of y, y' (we use (b) and (c». Thus, it is enough to prove (e) for a single x, namely, that given by Lemma 2.5. In this case, we set y = Tw" y' = Tw"Ta~1 , where w', w" are as in Lemma 2.5. Using Lemma 2.5, we are reduced to verifying the following identities: -I - (f) Taxsax' Tax = ~Ts(x) . -I - (g) Ta2X -oTax = T x-o . - -I (h) Taxs = Tx~ . Now Ts-1Taxsax = ~axsax (see Lemma 2.3(b) and (c» and saxsax = as(x)+x. Hence, (f) is equivalent to Tas(x)+x·T;;} = TS(x) which follows from the definition

of TS(x) and the fact that sex) +x = 2x - 2a E X dom (see Lemma 2.3(c». We

also have that x - a E X dom ' Hence, T X-o = Tax - o and (g) is equivalent to Tax' Tax-o = Ta2X -o, which follows from 1.4(g). Finally, (h) is equivalent to Taxs~ = Tax which follows from 1.4(dl). The lemma is proved.

Lemma 2.S. Let B' be the subgroup of B generated by Tx for all x E Xdom and by Tso for all a E II. Then B' = B . Proof. Let a E Ilrn' let A = (a, 1) E n, SA = soao E S (see 1.4). Now -a E X dom (since a E Ilm). We have aO(A) = aO(a, 1) = (a, -1) E R- . Hence, by 1.4(b), we have I(SAa-O) = l(a-O)-I. This implies Ta-o = TSATsAa-o = TSATso ' Hence, TSA = Ta-o Tso-I' E B . Thus, Ts E B , lorl" all s E S. Hence, Tw E B ' for all wEJJQQ,sincesuch w are of the form SIS2"'Sp (SiES, p=l(w».

Now let w E Q. We can find x E Xdom such that X· W E JJQQ and Txw E B' . We have TxTw = Txw' Hence, Tw = T;I . Txw E B'. Since B is generated by the elements Tw (w E Q) and Tw (w E WoQ) , we have B' = B.

3. THE HEeKE ALGEBRA 3.1. Consider the following three kinds of data: (a) a function L: W -+ N such that L(ww') = L(w) + L(w') whenever w, w' E W satisfy l(ww') = lew) + l(w') , (b) a function L 1: S -+ N such that LI (s) = LI (s') whenever s ,s' E S are conjugate in W, (c) a function A: II -+ N such that A( a) = A( a'}" whenever a, a' E II satisfy (a', a) = (a, a') = -1 , together with a function A*: {a E Ilia E 2Y} -+ N. We have a natural bijection between the set of functions as in (a) and the set of functions as in (b) and also a natural bijection between the set of functions as in (b) and the set of pairs of functions as in (c). The first bijection is defined

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by L-+LI = Lis, the second by LI-+(A,A*),where A(a)=LI(s,.) (aEll), A*(a)=LI(sa) (aEll, aE2Y, So as in 2.4). To see that it is a bijection, we use Lemma 1. 7. Hence, a datum of type (a) is equivalent to a datum of type (b) or one of type (c). Such a datum is called a parameter set for the root system. We shall assume that a parameter set for our root system has been fixed. Hence, L, L I , A, and A* are defined. We denote by l: B -+ Z the unique homomorphism such that l(Tw) = L(w) for all W E W . Many results in this section are due to Bernstein and Zelevinski, or Bernstein, in the "special case"mentioned in 0.1 (see [4, 4.3, 4.4]).

3.2. Let v be an indeterminate and let .SiI = C[ v , V -I]. The Hecke algebra H (over .SiI) is defined to be the quotient of the group algebra (over .SiI) of the braid group B by the two-sided ideal generated by the elements

SES.

The image of Tw E B (resp. Tx E B, see 2.6) in H is denoted again Tw (resp. Tx). It is well known that (a) the elements Tw E H (w E W) form a basis of H as an .SiI-module. We have the following identity in H: (b) (Ts + 1)(~ - V 2L(s)) = 0 (s E S). 3.3. For any x EX, we define (a) Ox = V-I(Tx)Tx E H. From Lemma 2.7(b) we see that (b) 0xOxl = 0X+XI for all x ,x' EX, 00 = 1. Lemma 3.4. (a) The elements TwOx E H (w E Wo' x E X) are linearly independent over .SiI . (b) The elements Ox Tw E H (w E ~, x E X) are linearly independent over .SiI. Proof. Assume that we have a relation E7=1 J; Tw; Ox; = 0, where (WI ,XI)' ... , (wn' xn) (n ~ 1) are distinct elements of ~ x X and 1;., ... ,In E.SiI - o. We can find x E X dom such that x + Xi E X dom for i = 1 , ... ,n. Multiplying our relation on the right by Ox' we obtain E7=1 J;Tw;Ox;+x = o. Hence,

Using 1.4(f), we deduce E7=1 J;v-L(ax+xilTw;Q::c+::c; = 0, contradicting 3.2(a). This proves (a). The proof of (b) is similar and will be omitted.

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3.5. Let (9 be the Jlf -submodule of H generated by the elements ()x (x E X). This is a subalgebra of H. From Lemma 3.4, we see that (a) Jlf[X] ~ (9, (x ~ ()x) is an Jlf -algebra isomorphism. (Jlf[X] is the group algebra of X over Jlf .) (b) Let K be the quotient field of the algebra (9. If x E X and 0: En, the element «()x - ()so(X))/(l- ()-a) E K actually belongs to (9. Indeed, it is equal to ()x(1 - ()~J/(1 - ()-a) , where n = (x ,0:). Similarly, if 0: E n is such that 0: E 2Y, then the element «()x - ()so(X))/(1 - ()-2a) E K actually belongs to (9. It is equal to ()x(l- ()~/2a)/(1- ()-2a), where n' = -!(x,O:) E Z. We can now state the following result.

Proposition 3.6. Let x EX, 0: En, s = Sa. We have the following identity in H: () - () ( Uta) _ 1) x s(x) v 1-()' i/O: ¢ 2Y, -a ()xTs -Ts()S(x) = «v2}.(a) _ 1) + ()_a(v}.(a)+}.*(a) _ v}.(a)-}.*(a))) ~x: ()()S(x) , -2a i/O: E 2Y. (Recall that ;'(0:) = L(s) , ;'*(0:) = L(s) , where s = sa is as in 2.4.) Proof. Assume that 0: is fixed and that the identity above is known for two elements x, x' of X. Then we see immediately that it also holds for x + x' and for -x. Hence, it is enough to prove the identity for x in a fixed set of generators of the abelian group X. If 0: ¢ 2Y , we can find XI E X such that (XI' 0:) = 1 , and X is generated by XI and by the elements x' E X such that (x', 0:) = o. If 0: E 2Y , we can find x 2 E X such that (X2' 0:) = 2, and X is generated by x2 and by the elements x' E X such that (x', 0:) = o. If {x ,0:) = 0, we have sx = x, and our identity reduces to ()xTs = ~()x which follows from Lemma 2.7(c). Therefore, it remains to prove our identity for x E X such that 1, if 0: ¢ 2Y, (x ,a) = { 2, if 0: E 2Y. Assume first that 0: ¢ 2Y and (x ,0:) = 1. From Lemma 2.7(d), we have Tx = ~TS(x)Ts in B. Applying L to this, we obtain L(T) = L(~(x))+2L(s). H ence, v L(tx)() x - T sV L(Ts(x))() s(x) T·s In H ,so tha t v 2L(s)() x T- s I - Ts () s(x). We substitute Ts- I = V-2L(S)Ts + (V-2L (S) - 1) (see 3.2(b)), and we obtain ()x~­ Ts()s(x) = (V 2L (s) - 1Wx which verifies the desired identity. Assume next that 0: E 2Y and (x ,0:) = 2. From Lemma 2.7(e), we have - I - I - -I 2 I I (a) TsTs(X) = yy , T x_a = yTsY ,TxTs = yTs y for some y, y E B. Applying L, we obtain (b) L(Ts(x))=vo-L(s), L(Tx_a)=vo+L(s), L(T)=vo+L(s)+2L(s), where Vo = L(y)- + L(y- I ).

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In the group algebra of B over .9J' we have

T T- I - (V2L(S) - I)T - V 2L(s)T T = (T: _ (V 2L(S) - I)T. _ V 2L(S)) I. x S X-a S s(x) )' S s)' The last expression has image equal to zero in H (see 3.2(b)). Hence, in H we have T x ~-I - (V 2L(S) - 1) T X-a - V2L(s) Ts Ts(x) = 0, or equivalently

L(Tx)() T- I -( 2L(S)_I) L(Tx- a )() _ 2L(s) L(Ts(x»)T() =0 v x s V V x-a V V s s(x) . Using (b), this becomes VL(s)+2L(S)() T- I _ ( 2L(s) _ 1) L(s)() _ 2L(S)-L(s)T () = 0 x s V V x-a V s s(x) . We again substitute Ts- I = v -2L(s) ~ + (v -2L(s) - 1) , and we obtain () T _ T () = ( L(s)+L(s) _ L(S)-L(S))() + ( 2L(s) _ 1)() x s s s(x) V V x-a V x which verifies the desired identity. This completes the proof of the proposition. Proposition 3.7. (a) The elements Tw()x E H (w E Wa, x E X) form an .9J' -basis for H. (b) The elements () x Tw E H (w E Wa, x E X) form an .9J' -basis for H. Proof. Let HI (resp. H2) be the .9J' -submodule of H generated by the ele- ments in (a) (resp. in (b)). Using Proposition 3.6 and induction on l(w), we see that Tw()x E H2, ()xTw E HI for any WE Wa, x EX. Hence, HI = H2 . Now HI is stable under left multiplication by elements Tw ' while H2 is stable under left mUltiplication by elements () x . Hence, HI = H2 is stable under left mUltiplication by elements Tw ' () x (w E Wa, x E X). But these elements generate H as an .9J' -algebra, by Lemma 2.8, and 1 E HI = H2 • It follows that HI = H2 = H. It remains to use Lemma 3.4. 3.8. We define for any Q E n an element ~(Q) E K (see 3.5(b)) by () v 2,l.(a) _ 1 a () _ 1 ' if Q ¢ 2Y , { ~(Q) = «()ava,l.(a)+,l.*(a) _ 1)«()av,l.(a)-,l.*(a) + 1) () _ 1 ' if Q E 2Y . 201 (This is reminiscent of the co-function in [6, p. 51].) We can reformulate the identity in Proposition 3.6 as follows. Proposition 3.9. In the setup of Proposition 3.6 we have ()x(Ts + 1) - (Ts + I)()S(x) = «()x - ()s(x))~(Q)· (The right-hand side is in tff.)

Corollary 3.10. If x EX, Q E n, then ()x + ()sQ(X) commutes with ~Q' Proof. We write the identity in Proposition 3.9 for x and for sa(x) and add the results. We see that ()x + ()sQ(X) commutes with ~Q + 1, hence the corollary. The following result is due to Bernstein, in the "special case" (see 0.1).

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Proposition 3.11. Let .z be the center of H. Then .z is the free .xi' -submodule of H with basis (zM = EXEM () x), where M runs over all Wa-orbits in X. Proof. From Corollary 3.10, we see that z M commutes with TSa for any a Ell. It clearly commutes with () x, for any x' EX. Hence, by 2.8, Z M is in .z. The linear independence of the elements Z M is clear from 3.5(a). The fact that they generate .z as an .xi' -module is proved by a specialization argument (reduction by v --+ 1 to the case of the group algebra C[W]) as in [3,8.1].

3.12. We have a natural .xi' -linear action w: f --+ w(f) of Wo on & such that w(()x) = ()w(X) for x EX, and the previous proposition implies that .z = &Wo (the Wa-invariants). This action extends to an action w: f --+ w(f) of Wo on K (by field automorphism). Let F be the quotient field of .z. We have .zc& n n FcK We have a natural isomorphism (a) & ®z F:::' K, e ® f ~ ef· (We must show only that, given any element eE & - 0, we can find e' E & - 0 such that ee' E.z. It is sufficient to take e' = IIwEwo ;w~1 w(e).) Let H F be the F -algebra H ® z F. It contains H as a .z-subalgebra. We identify the subspace & ®z F of HF with K using (a). Thus, we have KcHF • We have two decompositions (b) H = EBwEwo Tw& = EBwEwo &Tw (cf. Proposition 3.7). Tensoring with F over .z, we obtain two decomposi- tions, (c) HF = EBwEwo Tw K = EBwEwo KTw' In the F-algebra HF , we have (d) f(TSa + 1) - (TSa + l)sQ(f) = (f - sQ(f)):§(a) for any f E K, a E II. Indeed, by (a), we can write f = 1;/z, 1; E &, Z E.z - 0 and we are reduced to the case where f E& , in which case we may use Proposition 3.9.

3.13. For later reference, we define for any a E R: A.(a) = A.(a') , :§(a) = w(:§(a')) , where a' is a root in II and w E Wo is such that a = w(a'). (It is easy to see that this is independent of the choice of a' and w.) Similarly, for any a E R such that a E 2Y, we set A.*(a) = A.*(a') , where a' Ell, w E Wa, a = w(a'). Then the formula for :§(a) given in 3.8 remains true for any a E R. 3.14. Let!T be the torus Y ® C*. If x EX, we shall identify the basis element () x of & with the character () x: !T --+ C* , () x (y ® C) = C(x ,y) (y E Y , CE C*).

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Hence, we may identify the C-algebra & with the coordinate ring of the torus TxC· : to the basis element Vi () x of & corresponds the character TxC· -+ C* , (t, 'I) 1--+ ,: ()x (t). In particular, v corresponds to pr2: !T x C· -+ c* . Then K becomes the field of rational functions on !T xc* . We may also identify Y with the group of all algebraic homomorphisms c* -+ !T; to y E Y corresponds hy: C* -+!T, hy ({) = Y ® '" Now JJQ acts on !T by w: y ®, -+ w(y) ®, and on !T x C· by (t, 'I) -+ (w(t) , 'I)' This induces on functions the action of JJQ on & and K consid- ered in 3.12. We have (a) sa(t) = tho«()a(t»-I (0: E R, t E!T). We note also that if 0: E R , then (b) * if a ~ 2Y, ker(ho: C -+!T) = {{l}, {1, - 1}, if a E 2Y.

Lemma 3.15. Let t E !T. Then t is JJQ-invariant if and only iffor any 0: E TI we have (a) ()a(t) = 1, ifa~2Y, { ()a(t)=±1, ifaE2Y.

Proof. Condition (a) is equivalent to sa(t) = t (by 3.14(a) and (b», and it remains to use the fact that JJQ is generated by {salo: E TI}.

4. THE GRADED HEeKE ALGEBRA 4.1. We assume that a parameter set for the root system (X, Y ,R , R ,TI) has been fixed. In addition, we assume given a Wo-invariant element to E !T . To to corresponds a C-algebra homomorphism h: & -+ C defined by h(v) = 1, h«()x) = ()x(to) (x E X). By Lemma 3.15, we have for any 0: E TI, (a) 1, if ~ 2Y, { a h«()a)= ±1, ifaE2Y.

Let I be the kernel of h (a maximal ideal of &). Let &i = Ii /li+1 (i ~ 0), and let & = E9 i~O &i. This is a commutative graded C-algebra in a natural way. The action of JJQ on & induces an action of JJQ on & since I is Wo- stable (recall that to is JJQ-invarant). For any f E & , we denote by d(f) the image of f - h(f)1 in 1/12 = &1. We have (b) d(fl) = h(f)d(/) + h(/)d(f) for f ,I E & . If we regard H as a left &-module (see 3.12(b», we can consider the filtration (c) H ~ I H ~ 12 H ~ .... Lemma 4.2. The filtration 4.1 (c) is compatible with the multiplication in H, i.e., i H· I j He li+j H (i ,j ~ 0).

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Proof. We first show (a) Tw' Ii C Ii H for j ~ 0, W E UQ . We use induction on /(w). We see that it is enough to consider the case where w = sa. (0 E TI). We can also assume that j ~ 1. Using 3.12(d), we see that it is enough to check that for f Eli, we have (b) sa(f) Eli. (c) I, if Ii ¢ 2Y, f - Sa(f) i-I { ()C _ 1 E I , where c = a 2, if Ii E 2Y.

(d) ()a v 2J.(a) -1 E I if ~ ¢ 2Y. (e) ()av).(a)+h(Oa»).O(a) - h(()a) E I if Ii E 2Y . (Recall that if Ii E 2Y , we have h(()a) = ±1 and that the expression in (e) is one of the two factors of the numerator of ~(o) in 3.8.) Now (b), (d), and (e) are obvious. To verify (c), we note that II' - sa(II') = II' - Sa (I') + 1-sa(/) (I') ()C _ 1 ()C _ 1 ()C _ 1 sa a a a. which reduces us to the case where j = 1. We have for any f E & , (f - sa(f))/((): - 1) E & (see 3.5). Thus (a) is proved. It implies H· Ii eli H, where, in HIi , H is regarded as a right & -module. Similarly, we see that Ii H c H· Ii . Hence, (f) Ii H = HIi . We have, using (f),

The lemma is proved. 4.3. From the previous lemma, we see that - -i -i i i+1 (a) H = EBi>oH ,where H = I HII H inherits from H an associative C-algebra structure (the graded algebra asso- ciated to the filtration 4.1(c)). Moreover, & is naturally a subalgebra of H. (From 3.12(b), we see that l n l+1 H = Ii+1 .) Let tw be the image of Tw in HIIH = if (w E UQ), and let r = d(v) E &1 (see 4.1). Then tl is the unit element of H. Let K be the quotient field of &. Proposition 4.4. We have (a) H = EBwEWo &. tw = EBwEwo tw . &. (b) twtw' = tww' (w, w' E UQ). (c) ¢. tSa - tSa sa(¢) = (¢ - sa(¢))(g(o) - 1)('V¢ E &, 0 E TI), where

d(()a.) + U(o)r if d 2Y d(()) , 10'F , { g(o) = a. (g(o) E K) . d(()a) + (A,(o) +h(()a)A, (o))r if v 2Y d(()a) , I 0 E ,

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Proof. (a) follows easily from definitions and from 3.12(b). To prove (b) we may assume that W = sa (a E TI). We have

Now v 2).(a) - 1 E [. Taking images under H -If , we find (b). To prove (c), we note that ~ is generated as a C-algebra by ~I and that if (c) is true for ¢> and ¢>' in ~ , then it is also true for ¢>¢>'. Thus, it is enough to check (c) for ¢> E ~ or ~I • For ¢> E ~ , (c) is obvious. If ¢> E ~I , we can write ¢> = d(f) for some f E [ . Apply [H - [H/ [2 H = HI to the identity (d) fTSa - ~asa(f) = (f - sa(f))(~(a) - 1). The left-hand side of (d) is mapped to ¢>tsa - tsasa(f) , and it remains to show that the right-hand side of (d) (an element of ~ n [H = I) satisfies (e) d((f - sa(f))(~(a) - 1)) = (¢> - sa(¢>))(g(a) - 1). Let j=(f-sa(f))/(O~-I) (c asin4.2(c)). From4.2(c) we see that jE~. By 4.1(b), we have

¢> - sa(¢» = d(f - Sa (f)) = d(j(O: - 1)) = h(j)d(O: - 1) + d(j)h(O: - 1) = h(j)d(O:) = ch(hd(Oa)'

Hence, (f)

= cnh(hr = cnh(j)d(v - 1) = d(cnj(v - 1)),

where 2A(a) , if ¢ 2Y , { a n= A(a) + h(Oa)A*(a) , ifaE2Y.

Now nj(v - 1) E [, and from (f) we see that (e) is equivalent to the following statement: ~ 2 (g) (f - Sa(f))(~(a) - 1) - cnf(v - 1) E [ . Since f -sa(f) = j(O: -1) and j E ~ , we see that (g) would be a consequence of the following statement: (h) (0: - 1)· (~(a) - 1) - cn(v - 1) E [2 . Assume first that a ¢ 2Y . Then the left-hand side of (h) is

2 o (v n- 1) - n(v - 1) = (0 - l)(v n- 1) + (Vn--1 - 1) - n (v - 1) E [ . " a v-

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Assume next that a E 2Y. Then the left-hand side of (h) is 02a(v 2.l.(a) - 1) + Oa(v.l.(a)+.l.·(a) - v.l.(a)-.l.·(a») - 2n(v - 1) = (02a - 1)(v2.l.(a) - 1) + (Oa - h(Oa))(v.l.(a)+.l.·(a) _ v.l.(a)-.l.·(a»)

+ (v2~(~ ~ 1 _ 2A(a)) (v _ 1)

v.l.(a)+.l..(a) 1 ) + h(Oa) ( v-I -- (A(a) + A*(a)) (v - 1)

v.l.(a)-.l..(a) _ 1 2 -h(Oa) ( v-I -(A(a)-A*) (a)) (V-1)EI,

since 2n = 2A(a) + h(Oa)(A(a) + A*(a)) - h(Oa)(A(a) - A*(a)). Hence, (h) is verified and the proposition is proved. - - --Wo Proposition 4.5. Let % be the center of H. Then % = &' (WO-invariants in &'). The proof is the same as that of [5, 6.5]. 4.6. Let F be the quotient field of %. As in 3.12(a), we have (a) &'®zF:::'K. Let HE' be the F-algebra H ®z F. It contains H as a %-subalgebra. We identify the subspace &' ®y F of HE' with K using (a). Thus, we have K C HE'. Tensoring 4.4(a) with F over % , we obtain two decompositions (b) HE' = EBwEwo tw . K = EBwEWo K· tw. In the F -algebra HE" we have (c) ¢(tSa + 1) - (tSa + l)sa(¢) = (¢ - sa(¢))g(a) (¢ E K, a E TI). (Note that the WO-action on &' extends to a Wo-action on K by field automor- phisms.) This is deduced from Proposition 4.4(c) in the same way that 3.12(d) is deduced from 3.9.

5. THE ELEMENTS Tw AND t w

5.1. We preserve the setup of §§3 and 4. For any a E TI, we define Ta E HF ' t a E HE' by (a) Ta + 1 = (TSa + l)~(a)-1 , t a + 1 = (tSa + l)g(a)-1 .

Proposition 5.2. (a) There is a unique homomorphism T: WO -+ (group of units of HF) (resp. t: WO -+ (group of units of HE')) such that sa -+ Ta (resp. sa -+ ta) for all a E TI. (b) Let Tw = T(W), 'tw = t(w) (w E Wo). For any f E K (resp. ¢ E K), we have fTw = TwW- 1(f) in HF (resp. ¢tw = tww- I (¢) in HE'). The proof is based on the following lemma.

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Lemma 5.3. (a) Let cf> = EWEWo(-1/(W)v-2L(W)Tw E H and let h E HF be such that cf>fh = 0 (in HF) for all f E&'. Then h = O. (b) Let cf> = EwEWo(-l/(W)tw E H and let hE HE' be such that cf>¢h = 0 (in HE') for all ¢ E &'. Then h = 0 . Proof. Multiplying h (resp. h) by a suitable element of % (resp. %), we may assume that h E H (resp. h E H). We shall assume statements (c) and (d) below: (c) If hE H satisfies cf>fh E (v - l)H for all f E &', then hE (v - l)H. (d) If hE H satisfies cf>¢h E rH for all ¢ E &' , then hE rH. Using (c) and (d) and the assumptions of (a) and (b), we see that h = (v - l)h' (resp. h = rh') for some h' E H (resp. h' E H). But then (v - l)cf>fh' = 0 for all f E &' (resp. rcf>¢h' = 0 for all ¢ E &'). Hence, cf> fh' = 0 for all f E &' (resp. cf>¢h' = 0 for all ¢ E &'), and using (c) and (d) again, we see that h' E (v -l)H (resp. h' E rH), so that hE (v _1)2H (resp. hE r2H). Continuing in this way, we see that h E (v - l)i H (resp. h E ri H) for all i ;::: 1 . Hence, h = 0 and h = O. It remains to prove (c) and (d). First note that

for all f E &', a En, ¢ E &' (see Proposition 3.6 and Proposition 4.4(c)). Hence, by induction on l(w' ) , we have (e)

for all w' E Wo . Let h, h be as in (c) and (d). We can write uniquely h = EW/EWo TwJw" h = EW/EWo tWI¢WI (fW' E &' ,

¢w l E &'). See 3.12(b) and Proposition 4.4(a). By assumption, we have

'Lcf>fTwJw' E (v - l)H ('rtf E &'), WI

WI Hence, using (e), we have

WI

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Now TwTw' - Tww' E (v - I)H, V-2L(w) - 1 E (v - I)H for all w, w' E W. Hence, we deduce '"' l(w) ,-I L...t(-I) Tww'w (f)fw,E(V-l)H (VfE&),

W,w' • , 1/ or, settmg ww = w , '"' l(w'w") ,-I L...t (-1) Twllw (f)fw' E (v - I)H ('If E &), w',W" '"' l(w'w") ,-I - - L...t (-1) tWIIW (¢J)¢Jw,ErH (V¢JE&). w',w" Using 3.12(b) and Proposition 4.4(a), we deduce that (f) Ew,(-I)I(W')W,-1 (f)fw' E (v - 1)& ('If E &), (g) Ew,(-I)I(w')w,-1 (¢J)¢Jw' E r& (V¢J E &). We write (f) (resp. (g)) for f = O~, i = 0,1, ... ,1»01- 1 (resp. ¢J = ¢J~, i = 0, 1 , ... ,1»01 - 1), where x E X (resp. ¢J, E &') is fixed such that x f. w(x) (resp. ¢J, f. w(¢J,)) for all w E Wo. Using Cramer's rule, we see that for all w' E »0 ' we have (h) ofw' E (v - 1)& (resp. (5¢Jw' E r&), where 0 is a Vandermonde determinant in the variables w-'(Ox)' w E »0 (resp. w-'(¢J,), WE »0). Hence, 0 (resp. (5) is a product of elements of &

(resp. &) of the form w~' (Ox)-w;' (Ox)' w, f. w 2 (resp. w~' (¢J, )-w;' (¢J,) , w, f. w2 ). These factors of 0 (resp. (5) are nonzero and are not divisible by v-I (resp. r). Since & (resp. &) is a unique factorization domain, from (h) it follows that fw' E (v - 1)& (resp. ¢Jw' E r&) for all w' E »0, and (c) and (d) follow. The lemma is proved. 5.4. Proof of Proposition 5.2. If cI> is as in 4.3(a), we have clearly (a) cI>(Tso + 1) = 0 for any a En. If fEK, aEn, we have (in HF )

cI>f(ro + 1) = cI>f(~o + 1)~(a)-' = cI>((Tso + l)s)f) + (f - so(f))~(a))~(a)-' (see 3.12(d)) = cI>(f - So (f)) (see (a)).

Hence, cI>fro = -cI>so(f). Applying this identity repeatedly, we get (b) cI>frolr02···roP=(-I)PcI>s ···s s (f) Op 02 01 for any sequence a" a ' •.• ,a in n. Assume now that s s ... s = I in 2 p UJ 02 cx.p Wo. Then p must be even and we also have s ... s s = 1 in Wo. From Q p 0'2 01

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(b), we deduce that .al .a2 ••••ap = 1. This proves Proposition 5.2(a) for •. We now rewrite 3.12(d) using 5.1(a): f(.a + l)~(a) - (.a + l)~(a)sa(f) = (f - sa(f))~(a). Cancelling ~(a), we obtain f.a = .asa(f) (f E K, a E TI) and Proposition 5.2(b) follows (for .). The proof for t is completely similar.

Proposition 5.5. (a) HF = EBwEwo .w • K = EBwEwo K . • w . (b) Hp = EBwEWo twK = EBwEwo Ktw' Proof. Let H~ = EWEWo .w·K. We show by induction on /(w') that Tw' C H~ for any w' E Wo' It is enough to show that ~a H~ C H~ for any a E TI or, using 5.1(a), that .aH~ C H~ and KH~ C Hp. Now .aH~ C H~ follows from the definition of .w and KH~ C H~ follows from Proposition 5.2(b). Thus, Tw' C H~ for all w' E Wo' Hence, Ew' Tw,K c H~. Hence H~ = HF (see 3.12(c)). Now Tw (w E ~) form a basis of HF as a right K-vector space, and we have just seen that.w (w E ~) form a set of generators of HF as a right K-vector space. It follows that .w (w E Wo) form a basis of this vector space. Hence, HF = EBwEwo .w . K. The equality HF = EBwEWo K . • w and the equalities in (b) are proved similarly.

6. SOME BRAID GROUP RELATIONS 6.1. This section contains a result which is needed in the proof of Theorem 8.6. We fix a "I- P in TI. Let ~aP be the subgroup of ~ generated by sa' sP' and let m be the order of the product sasp' We assume given a finite set g on which ~aP acts by permutations and an element C E g. We define elements C;1 ,C;2' .. , ,111,112' ... in HF by if sa(c) = c, if Sp(c) = c,

if sa(c) "I- c, if S p ( C) "I- C , if spsa(c) = Sa(C) , if SaSP(C) = Sp(C) , if spsa(c) "I- Sa(C) , if SaSP(C) "I- Sp(C) , if Sa Sp Sa (c) = SPSa(C) ' if SPSaSP(C) = SaSp(C) , if SaSPSa(C) "I- SpSa(C) , if Sp Sa Sp (C) "I- SaSp(C) , etc.

Proposition 6.2. We have C;1C;2" 'C;m = 111112" . 11m in HF ·

Proof. Let Z be the set of reflections in ~aP which keep C fixed. To simplify notation, we set .a = ., .p = .', TSa = T, TS{I = T' . Assume first that Z is empty. Then the identity to be proved is (a) H'.,,· = .'H',,· (m factors). This is known from Proposition 5.2(a). We denote by i the two sides of (a).

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When Z contains m reflections, the identity to be proved is (b) TT'T··· = T'TT'··· which follows from the definition of H. Assume now that Z = {s,,}. The identity to be proved is (c) Tr' = r'T (if m = 2), Tr'r = r'rT' (if m = 3), Tr'rr' = r'rr'T (if m = 4), Tr'rr'rr' = r'rr'rr'T (if m = 6). A simple computation using Proposition 5.2(a) and (b) shows that both sides of (c) are equal to r(1 + r P(1 - ~(p)-I»~(p), if m = 3, { r( 1 + r a (1 - ~ (a) -I»~ (a) , if m = 2 , 4, or 6 . The case where Z = {sp} is entirely similar. We can assume from now on that m;?: 3. Assume that Z = {saspsa}' The identity to be proved is (d) rT'r = r'Tr' (if m = 3), rT'rr' = r'rT'r (if m = 4), rT'rr'rr' = r'rr'rT'r (if m = 6). This can be formally deduced from (c) using the fact that r2 = l = 1 . More generally, the case where Z consists of a single reflection follows formally from (c). We can assume from now on that m ;?: 4. When m = 4, it remains to consider the cases where Z = {sp ,saspsa} and Z = {sa ,spsasp}, We can assume that (a, /1) = -1, (P, a) = -2. The identities to be proved are, respectively, (e) rT'rT' = T'rT'r, r'Tr'T = Tr'Tr'. A simple computation using Proposition 5.2(a) and (b) shows that both sides of the first identity of (e) are equal to r(l + rP(1 - ~(p)-I) + ralra(1 _ ~(2a + p)-I) + i(1 - ~(P)-I)(1 - ~(2a + P)-I»~(p)~(2a + P) and both sides of the second identity of (e) are equal to r(1 + ra(1 - ~(a)-I) + l rarP(1 - ~(a + p)-I) + r(1 - ~(a)-I)(1 - ~(a + p)-I»~(a)~(a + P); we have ~(2a + P) = sa(~(P», ~(a + P) = sp(~(a», sp(~(2a + P» = ~(2a + P), sa(~(a + P» = ~(a + P) (see 3.13). When m = 6, we can assume that (a, /1) = -1, (P, a) = -3. It remains to consider the cases where Z = {sa ,spsaspsasp} , Z = {sp ,saspsaspsa} ' Z = {saspsa ,spsasp} , Z = {sp ,saspsa ,spsaspsasp} , Z = {sa ,SpSaSp ,saspsaspsa}' The identities to be proved are, respectively, (f) Tr , rT "rr = r , rT "rr T, rr'" Tr rT = T '"rr Tr r, rT'" rr Tr = r 'T r ' r T' r, rT'rT'rT' = T'rT'rT'r, r'Tr'Tr'T = Tr'Tr'Tr'.

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These can again be verified using Proposition 5.2(a) and (b). We omit the details. (Note only that the first three identities in (f) are formally equivalent to each other.)

7. COMPLETIONS 7.1. We preserve the setup of §§3 and 4. Let t = Y ®C. Any element x E X may be regarded as a linear form t ---- C, y®z ---- z(x ,y). This linear form will be denoted again by x. We shall denote by x the composition tEBC~t~C. There is a unique isomorphism Hom(tEBC,C) ~ ~1 = 1112 (see 4.1) such that x 1--+ d(Ox) , pr2 1--+ ,. We shall identify these two vector spaces. In particular, we shall identify , with pr2 : t ® C ---- C. This identification is compatible with the natural Wo-actions. It follows that ~ may be identified with the algebra of regular functions t EB C ---- C . 7.2. By Proposition 3.11 (resp. Proposition 4.5), % = ~Wo (resp. % = ~Wo) may be identified with the coordinate ring of (g- x C*)IWo = (g-I»Q) x C* (resp. (t EB c)/»Q = (tIWo) x C) so that the inclusion % c ~ (resp. % c ~) corresponds to the orbit map g-xc* ---- (g-xC*)/»Q (resp. tEBC ---- (tEBc)/»Q). Hence, the maximal ideals of % (resp. %) are of the form (a) J(r.,vo) = {f E %If(t, vo) = 0, 'tItE1:} (resp. '(f.,ro) = {¢ E %I¢(c; "0) = 0, 'riC; E ~} , where 1: (resp. ~) is a »Q-orbit in g- (resp. t) and Vo E C* (resp. '0 E C) .

?y. We now fix (1:, vo) (resp. (~, '0)) as in 7.2. We denote by % (resp. %) the J(r.,vofadic (resp. '(f.,rofadic) completion of % (resp. %). We define

~ ~ &=~®%, ~=~®%, H=H®%, H=H®%.- z z z z

~ ~ Then &, H (resp. ~, H) are naturally%- (resp. %-)- algebras and the imbeddings % c ~ c H (resp. % c ~ c H) give rise to imbeddings !Z c ~ ~ & c H (resp. %- c ~ c H). We shall regard ~ and H as %-subalgebras of t! and iJ in the obvious way. We also regard ~ and H as Z-subalgebras of ~ and H. We note that the %-l~ear (resp. %-linear) »Q-action o~ ~ (resp. ~) extends to a %-linear (resp. %-linear) »Q-action on & (resp. ~), and we have .-. _ .::::::::::. ~Wo (a) % = ~wo , % = ~ . Let F (resp. F) be the quotient field of % (resp. %). Let K (resp. ~) b: th: fult... ring of quotients of & (resp. ~). We have FCK, &CK, FcK, ~cK.

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Just as in 3.12(a), w: see u~ing ~a) that (b) tff®!iF ~ K, &®;F ~ K. This is defined by multiplication in K or K. Using the definition of tff, we obtain K 2:: & ®z F 2:: & ®z F ®F F. Hence, (c) K2::K®FF (see 3.1}(a)). Sim~arly, (d) K 2:: K ®pF.

~ 7.4. W~ consider the~ F-algebra ~ = H ®z F = HF ®F F (resp. the F- algebra HE = H ®Z F = HE ®E F). We can naturally regard HF (resp. HE) as an F-subalgebra of HF (resp. F-subalgebra of HE). Moreover, the imbedding K C HF (see 3.1:' resp. K c:. HE .isee 4.6)) induces an imbedding K = K ®F F C HF (resp. K = K ®F F c HE). From 3.12(c), 4.6(b), and Proposition 5.5, we deduce that (a)

From 3.12(b) and Proposition 4.4(a), we deduce that (b) { ! = ffi Tw~ = ffi~Tw ' H = ffitw& = ffi&tw (all direct sums are taken over W E UO) . 7.5. For any tE1: and l E f, we define I(tl = {f E &If(t ,vo) = O}, a maximal ideal of & , I(T) = {¢> E &I¢>(l, '0) = O} , a maximal ideal of & ,

~ = I(ll-adic completion of &,

&[ = I(T)-adic completion of &.

We have J(r.,vol C I(tl (resp. l(I.,rol c I(T)). Hence, the identity map & -- & (resp. & --~&) e~tends continuously to a homomorphism of completions tff-- ~ (resp. & -- &1). Taking the direct sum over t (resp. l), we obtain an isomorphism of %-algebras (a) tff ~ ffilEr. ~ (resp. aE- isomorp~sm of %-algebras (b) & ~ ffiIEI.&I). (In a direct sum of algebras, the product of two elements in different summands is defined to be zero.)

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The natural action of WQ on & (resp. &) (see 7.3) corresponds under (a) and (b) to a WQ-action which permutes the summands ~ (resp. &/) according to the transitive Wo-action on 1: (resp. !).

7.6. In the setup of 7.5, let &an (resp. &an) be the algebra of holomorphic functions !T x C· -> C (resp. t E9 C -> C), let It') (resp. I ~~) be its max- ~ ~an imal ideal defined by (t,vo) (resp. (l"o)) , and let &:n (resp. &t ) be the corresponding It') -adic (resp. I ~~ -adic) completion. It is clear that we have natural isomorphisms - ~ -an -,...... , -an (a) ~ = & t ,&/ = & / .

8. FIRST REDUCTION THEOREM 8.1. In this section, we preserve the setup of §3. We assume given a WQ-orbit 1: in !T and an element Vo E c* . Let (vo) be the subgroup of C· generated by vo' and let !T (vo) be the subgroup Y ® (vo) of Y ® C· = !T. Clearly, !T (vo) is WQ-stable. For any t E !T , we define

R = {O:ER/oa(t)E(Vo), if 0: ¢ 2Y } t 0a(t)E±(Vo)' if 0: E 2Y , Rt = {O: E RIo: E R t }, + + Rt =RtnR , ~ + TIt = set 0 f all 0: E R+.t WhICh are not 0 f the form 0:'". + 0: WIth 0:'" ,0: E R t ' t WQ = subgroup of Wo generated by the sa (0: E Rt ),

~t WQ = {w E WQlw(t) = t}, ~t + + rt = {w E WQlw(R t ) =Rt }.

Note that (X, Y ,Rt ,Rt ,TIt) is a root system. (We must check that 0:, P E Rt => sp(O:) E Rt • We have

-1 -(a P) Osp(a)(t) = 0a(sp(t)) = 0a(t· hp(Op(t)) ) = 0a(t)Op(t) , .

We have Op(t)-(a'P)E(Vo) since PERt. Hence, 0S{J(a)(t)EOa(t)·(vo) andour assertion follows.) Note also that TIt rt TI in general. Clearly, W~ is a normal ~t subgroup of WQ with complement rt • Note that (a) R t , Rt , ••• ,rt depend only on the !T (vo)-coset of t, not on t itself. We define an equivalence relation on 1: as follows. We say that t, t' E 1: are equivalent if t, t' are in the same !T(vo)-coset and t' = w(t) for some t t' WEWQ=WQ

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Let fJiJ be the set of equivalence classes. It is clear that if t, t' E ~ are equivalent and w E Wo' then w(t), W(t') are equivalent. Hence, Wa permutes (transitively) the sets in fJiJ. For C E fJiJ , let

Wa(c) = {w E Walwc = c} be the isotropy group of c. If c E fJiJ , we shall write Rc' Rc' R;, IIc' W;, W;, rc instead of R t , n n+ t ~t C ~c .fit' .fit ' IIt , Wa, Wa, r t for t E c (see 8.I(a)). Clearly, Wa c Wa(c) c Wa . Let r(c) = Wa(c) nrc' Then Wo(c) is a semidirect product of r(c) and the normal subgroup W;.

Lemma 8.2. (a) If Yo is the subgroup of Y generated by R, then we have t E ~, w E W~ * w(t)t- I E image(Yo ® (vo) .t Y ® C*) (r/J is the natural map). (b)IfaER, cEfJiJ, then aERc¢}s,,(c)=c. Proof. We prove (a). Since Yo is stable under Wa, we may assume that w = sp

(P E R t ). We have sp(t)t-I = hp«()p(t)-I) = hp(vv;), where n E Z and I, if ¢ 2Y, v- { /J ± 1, if /J E 2Y. If /J E 2Y, we have hp( -I) = 1. Hence, in any case, hp(v) = 1 and sp(t)t-I = hp(-l) = /J®v;, as required. We now prove (b). The implication * is obvious. We prove the converse. Assume that s,,(c) = c and choose t E c. Then t ,s,,(t) are in the same W~­ orbit, i.e., s,,(t) = w(t) for some w E W~. Using (a), we have S,,(t)-I = w(t)· t-I E image(r/J). We set u" = (),,(t)-I E C* . Then a ® u" E image¢. In particular, a ® u" E Y ® (vo)' If a ¢ 2Y , then a is not divisible in Y and it follows that u" E (vo) , i.e., a E R t • Assume now that a E 2Y. Using Lemma 1.7, we can assume that our root system is primitive. By properties of root systems of type C, we see that we can number the roots in II as a l ,··· , an such that a = a l +c2a 2 +c3a 3 + ... +cnan (c2 , ••• , cn are integers ~ 0), where a l E2Y and !al ,a2 ,a3 , ••• ,an form a basis of Y. From a®u"Eimager/J, we have (writing now the operation in Y ® C* as addition), n n a I ® u + ~ c.Q. ® u = ~ a. ® vd; (d E Z). Q ~ll a L..-Jl o i i=2 i=1 Hence, • n • n ~,o, 2 + ~. ,0, c; _ a 1,0, 2dl + ~. ,0, d; 2 '01 U" L...J a i '01 U" - 2 '01 Vo L...J a i '01 Vo . b2 b2

Since a)2, a 2 , ... , an form a basis of Y, it follows that u: = v;dl , so that dl . U" = ± Vo and a E R t • The lemma IS proved.

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8.3. For c E g, we denote by He the Heeke algebra defined in terms of the root system (X, Y ,Re ,Re, lle) and the parameter set (see 3.1(c)),

A.e(a) = A.(a) (a E lle) , A.;(a) = A.*(a) (a E lle' it E 2Y)

(A.(a) , A.*(a) as in 3.13) in the same way that H was defined in terms of (X, Y ,R, R, ll) and the parameter set A., A.* . Since r(c) acts on (X, Y ,Re , Re , lle) compatibly with the parameter set (since r(c) cWo), it also acts naturally on the algebra He: if Tw /}x (w E W; , X E X) are the basis elements of He analogous to the basis elements TJ}x (w E »0, X E X) of Hand y E r(c), we have y(Tw,e(}x) = Tywy - I ,/ly(X).

8.4. Consider the maximal ideal JCr.,vo) of .:z (see 7.2(a)). Similarly, for c E 9 , we consider the maximal ideal

of ~ = &,Wc,c = center(He). (Note that c is a W; -orbit in Y.) In 7.3, we introduced the 'tl:,vofadic completions !Z, tff, H of the %-algebras % , &' , H. Similarly, let ~, ~, He be the J(e ,vofadic completions of the ~ -algebras ~,&', He· We shall also need K, HF of 7.3 and 7.4. The action of r( c) on He extends continuously to an action of r( c) on He' since r(c) leaves stable the maximal ideal J(C,vo) of ~. (Recall that r(c) c Wo(c) .)

8.5. If A is an associative ring with 1, denote by An the ring of all n x n matrices with entries in A. If a finite group r acts on A by ring automor- phisms, we can define formally a new ring A[r] = ffiyEr A·y with multiplication (ay)(a'y') = (ay(a')).(yy'). (The group algebra of r over a field is a special case of this.) In particular, the action of r(c) on He gives rise to a ring HJr(c)]. This is not, in general, a ~-algebra since r(c) may act nontrivially on ~. It is only a !2[(C) -algebra. :Cr( ) ~w.c f( ) ~w. ( ) ~ ~w. ( ) We have ~ c = (~ 0) c = &'e 0 c . Thus, Hc[r(c)) is a ~ 0 c -algebra.

The identity map &' ---t &' extends continuously to a ring homomorphism tff ---t ~ (since J(l:,vo) c J(c,vo)). This restricts to a ring homomorphism i: tffwo ---t ~wo(C) (since Wo(c) cWo). It is clear that i is an isomorphism. (By 7.5(a) and its analogue for ~, both tffWo and ~Wo(e) are isomorphic to the ring of invariants on ~ (for some tEe) with respect to the stabilizer of t in »0 (or in Wo(c)). Via i, we can regard Hc[r(c)) also as a tffwo-algebra.

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We can now state

Theorem 8.6. If c E .9, there exists an isomorphism of &WO -algebras H ~ HJr(c)]n' where n = #.9. The proof will occupy most of this section.

8.7. Recall the decomposition & = EBtE1:~ (in 7.5(a)). For any t E~, we denote by I t the unit element of ~. We also regard I t as an element of &.

Then the unit element I of & and of H satisfies I = EtE1: I l' It' I t' = ~t ,t' . It (t,t'E~) and w(lt)=lw(t) (WEWo, tE~). Let c E.9. We define (a) Ic=EtEcltE&cH. We may identify ~ (see 8.5) with the subring lc& = &le = EBtEe~ in the obvious way. It is clear that (b) I = EcE.9' Ie' lc' ICI = ~e,cl . Ie (c, c' E.9), w(1c) = IW(C) (w E UO, c E.9). For c, c' E.9 , we define (c) cHcl = IcHlcl C H. From (b), we see that .- ...... - -- . , (d) ~ = EBcE.9'~' ~~-- C ~, ~~, = If c -I c ...... ° 0" (e) H = EBc,c/E.9' cHel' eHel' elHclI C cHell' cHcl . c;HclI = °If c -I C1 • 8.8. Let 0: En, c E.9. We define an element ~ E HF by (a) r; = { le~o = ~s)sJC), ~f 0: E Re (~.e., ~f s,,(c) = c), o lcr = r IsJc) , If 0: ¢ Re (1.e., If s,,(c) -I c) (we use s,,(lc) = Iso(C) and Lemma 8.2(b)).

Lemma 8.9. (a) If 0: En, 0: ¢ Rc' then the rational function ~(o:) on !T is regular and nonzero at all points of c U s,,(c). (b) We have ~ E cHso(e) for all 0: En. Proof. Let 0: En, 0: ¢ Rc' and let t E c. We show that the numerator and denominator of ~(o:) (see 3.8) do not vanish at t and s,,(t) , i.e., O,,(t)±lV~;'(") - I -1O, O,,(t)±l - I -10, if a ¢ 2Y, { O,,(t)±lV~(")+;'·(") - I -I 0, O,,(t)±lv~(,,)-;'·(a) + I -1O, O;2(t) - I -1O, if a E 2Y.

(We have 0a(sa(t)) = 0a(t)-l.) But this follows from 0: ¢ Re' This proves (a). Now (b) is clear if 0: E Re' Assume now that 0: ¢ Re' We have ~ = lerQIsn(e) = le(r" + l)lso(e) (since lclsa(e) = 0). Hence, using 5.I(a), ~ =

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le(TSa + 1)~(Gt)-IISa(e)' From (a), we see that ~(Gt)-IISa(e) E ~a(e) and (b) follows. 8.10. Given c E.9 and W E »0 ' we define (a) r = T e ral(e) ... rap'''Sa2Sal(e) W Sal Sal Sap , where Gt l ,Gt2 , ... ,Gtp is any sequence in II such that w = salsa2" . sap , p = l(w) . From Lemma 8.9 and 8.7(e), we see that (b) r:., E /lw-I(e) • Proposition 8.11. The element r:., is well defined (it is independent ofthe choice of Gt l ,Gt2 , ... ,Gtp )' Proof. For any Gt E II, we define ~a=~'(EH. eEfJIJ We first show that for any Gt =F P in II such that sasp has order m, we have = (a) TSa TSp TSa ... TSp TSa TSp ... (both products have m factors). This is equivalent to the identity '"' (T ra(e) rpSa(e) ..• ) = '"' (T rp(e) raS(J(e) ..• ) , L..J Sa Sp Sa L..J S(J Sa Sp eEfJIJ eEfJIJ where all products have m factors. (We have used Lemma 8.9 and 8.7(e)). Therefore, to prove (a) it is enough to show that for any fixed c E.9 , we have r ra(e) rpSa(e) .•. = T rp(e) raSp (e) ...• Sa Sp Sa Sp Sa Sp Using the definition 8.7(a), we see that this is equivalent to 1eele2 .. · em = 1e171172" . 17m ' where e;, 17; are as in Proposition 6.2. (We have used 8.7(b).) This is a consequence of Proposition 6.2. Thus, (a) is proved. By a well-known property of the braid group of Wo' we see from (a) that we can define for W E »0 ~ ~ ~ E Tw = al Q2 ... ap H,

where Gt l ,Gt2 , ••• ,Gtp is any sequence in II such that W = salsa2" . sap , l(w) = p (and this is independent of the choice). Then r:., as defined in 8.1O(a) is the projection of Tw onto the eHW-1(e(summand in the decomposition H = E9 eHel' Hence, it is intrinsically defined. This completes the proof of the proposition.

Lemma 8.12. Let c E .9, and let Gt E IIe be such that l(sa) > 1 (length in Wo)' Then (a) there exists P E II such that l(spsasp) = l(sa) - 2. (b) If P is as in (a), then P ¢ R; .

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Proof. (Compare [2].) Being a reflection, sa has odd length. Hence, [(sa) ~ 3. Let P E TI be such that /(spsa) = /(sa) -1. Then /(sasp) = /(spsa) = /(sa) -1. We can find a sequence 0: , ••• ,O:p in TI such that S = sps S ···s ,/(s~) = 1 a at ll2 Up lA P + 1. Since [(sasp) = [(sa) - 1 , we see from the exchange condition that either there exists m ~ 1 with

or Sp SaI Sa2·· . sap = SaI Sa2·· ·sapsp. In the first case, we have S = sps ···s S S ···s sp and (a) follows. a 01 Om-I Om+l ll'm+2 o.p In the second case, we have S S S ... S sp sps sp Ss ( ). Hence, a = al a2 ap = a = fJ a 0: ,sp(o:) are proportional. Hence, sp(o:) = ±o:. We cannot have sp(o:) = -0:: the only positive root taken by sp to a negative one is P (recall that 0: ¢ TI, so 0: '# P). Hence, sp(o:) = 0:, so (0:, /3) = o. But then (P, a) = o. Hence, sa(P) = P E R+ , so /(sasp) = /(sa) + 1, a contradiction. Thus, (a) is proved. We now prove (b). Assume that PER; . From [(sasp) = /(sa) - 1, it follows that sa(P) E R- . Thus, sa carries two nonproportional roots in R; (namely, 0: and P) to negative ones. This contradicts the fact that 0: E TIc. The lemma is proved.

Lemma 8.13. Let c E 9', Y E r(c), and let 0: 1 ,0:2 , ••• ,O:p E TI be such that y S S ... S ,[(y) p. Then = al a2 ap = (a) c '# sal (c) '# Sa2 Sa l (c) '# ... '# Sap· .. Sa2 Sa l (c) (i.e., 0: 1 ¢ Rc' Sal (0:2) ¢

Rc' ... 'Sal Sa2 ... Sap_1 (O:p)¢Rc). (b) r; = 1c -r a I -ra2 ••• -rap 1c . (c) r; ~ = r;w' T~ T; = ~Y' for all W E W; . (d) r;f1c= lcy(f)r;,forall fE&. Proof. We first note (e) if y E Wa, P E TI, sp(c) '# c, then -rpr; = r::;C). Indeed, if /(sPy) = /(y) + 1, this follows from definitions (8.8 and 8.10). If /(sPy) = /(y) - 1, the same definitions show that r; = -rPT::;C). Multiplying on both sides by -rP and using -rP-r P = 1 , we again find (e). E E Assume now that we have J = Sat S02 ... S.0,_1 (0:,.) Rc for some i [1 ,p]. Since [(Sal··· sa) = i, we have J E R+. Hence, J E R;. We have O:j = S ···s (P). Now S ···s (0:.) E R+ since [(s ···s 0:.) = p - 1. Ui-I at Up 0';+1 I Up 0;+1 I i + Hence, y-1(J) S ···s S (J) S ···s S (0:.) -s ···s (0:.) E R- . = o.p 02 0:1 = up Ui+l Uj I = up 0;+1 I By assumption, y E rc. Hence, y(R;) = R; . This contradicts y-1(J) E R- . This contradiction proves (a). Now (b) follows from (a) using the definitions 8.8 and 8.10. Using (e) repeatedly, we see that (c) follows from (b). Using the

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identity frO. = rasa(f) in HF (f E &, 0: E II), we see that (d) follows from (b). The lemma is proved.

Lemma 8.14. Let c E .9, 0: E IIe . (a) We have (r;, + 1)(r;, - v 2A(a)) = 0 in H (,1,(a) as in 3.13). (b) If WE W; is such that w-I(o:) E R;, then T~ r:v = r;,w' (c) If fE&, then fr;, = ~sa(f)+ 1e(f-sa(f))(~(0:)-1) in H.

Proof. We argue by induction of [(sa)' If [(So.) = 1, we have ~ = 1eTsa' Hence, (a) and (c) follow from 3.2(b) and 3.12(d); (b) follows from 8.8(a) and 8.10(a). Now assume that [(So.) > 1. Choose P E II so that 0:' = sp(o:) satisfies [(sa') = [(So.) - 2 (see Lemma 8.12(a)). Set c' = sp(c). By Lemma 8.12(b), we have P ¢ R; . Hence, sp(R;) C R+ . (Recall that the only positive root taken by s p to a negative one is p .) Hence, sp(R;) = R; and sp(IIe) = IIe' . In particular, 0:' E IIe" We may assume that the lemma is true for (0:', c') instead of (0:, c) , since [(sa') < [(S,,) . We have (d) y-c rPY-C' r P Sa = so' (using c =F sp(c) = c' which follows from Lemma 8.12(b) and using 8.10(a)). Now (a) and (c) follow from the corresponding identities for r::, using (d) p p 'e' and r r = 1. We now prove (b). Let W = spwsp E Wo . We have ,-I, -I' -I + + W (0:) = spw sp(o:) = sp(w (0:)) C sp(Re) = Re" Hence, r::, T~, = r::,w' , using the induction hypothesis. From the definition 8.10(a), we see, using c =F sp(c) = c' (just as in 8.13(e)), that e p,."c' p Tw=r 1w,r . We now compute P'p p'p P"p pc' pc ~~ = (r ~,r )(r ~,r ) = r ~,~,r = r Tsa,w,r = TSaW' The lemma is proved.

Lemma 8.15. Let c, c' E.9. Then eHe' = EBwEwo ;w(e)=e' (~ . r:v) . Proof. We shall prove by induction on [(w) that (a) 1eTw Ie' C /ie, ('Vw E Wo) , where (b) /Ie, = EW'EWo;W'(e)=e'~r:v,. When W = 1, (a) is trivial. Hence, we may assume that W = saw) , where 0: E II, [(w) = [(WI) + 1 and that (a) is known for w) instead of w. Assume first that 0: E Re' Then

1e Tw Ie' = 1e TsaTWl l e' = TSa 1e Tw I Ie' E TSaeHe' .

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We have Tso Ii c Ii~o + Ii. Hence, Tso ~ c ~Tso + ~ and leTw leI e EWI ~~o ~, + EWI ~~, (w' (e) = e' in the summation). If I(saw') = I(w') + 1 , we have ~o ~, = r:"wl' If I(saw') = I(w') - 1, we have ~o ~, = r:" r:" r:"wl ': (v 2A(a) - l)~, + v2.i(a)r:"wl (see Lemma 8.14), and we see that leTw leI C eHel . Assume next that a ¢ Re' Then leTw leI = leTs" TWllel = lira~(a) + (~(a) - l))Twllel = ra~(a)lsJe)Twllel + (~(a) - l)leTwllel e ra$(a)sa(e)Hel + (~(a) - l)cHcl' Now ~(a) is regular at all tee U Sa (e) (see Lemma 8.9(a)). Hence, ~(a) can be absorbed in so(e)Hcl and Hc,el' Thus, lcTwlcl e raso(c)Hel + cHe" It remains to show that raSo(c)Hc' C cHe' . This follows from the equality 8.13(e). Thus, (a) is proved. We now show that the sum defining eHc' in (b) is direct. It is enough to show (c) a relation EW'EWo fw'~' = 0 (fw' e~) implies that all fw' are zero. Assume that there exists a relation as above with not all fw' equal to zero. Let 10 be the maximum length of an element such that fWI t- O. From the definition (8.10) of ~, we have that

~, = Ie E Yw' ,w"Tw'" w"EWo where Yw' wIt e K are such that Yw' wIt = 0 unless w " ~ w , in the Bruhat order and 'Yw' ,W' t- O. Hence, from (c) it follows that (d) E W'EWo;l(w'):5!o lw,Twl = 0 (in HF ), where lw' e K are such that lw' = fw' . Yw' ,w' for I(w') = 10 , The product in K of a nonzero element of ~ with a nonzero element of K is nonzero, since the natural homomorphism ~ --+ ~ is injective and ~ is an integral domain for any tee. It follows that lw' t- 0 for some w' with I(w') = 10 and, therefore, (d) contradicts 7.4(a). This contradiction proves (c). The proposition is proved. 8.16. Proof of Theorem 8.6. Recall that e e.9 is fixed. For any e' e.9, we choose a sequence ,(e')={aI ,a2, ... ,ap } in n such

that e t- Sap (e) t- Sap_1 Sap (e) t- ... t- Sal Sa2 .. ,sap (e) = e' . I II - - For any e ,e e.9, we define ae, ,cIt : cHc --+ c,Hc" by

A al a2 p a c, ,cIt (h) = r r ... r aPh r P' r Pp'-I ... r PI

= Cr' TOI(c') ... )h(r T:llp,(C) ... ), SOl S02 slip' IIp'_1

where '(e') = {al ,a2 , ... ,ap }, ,(e") = {PI ,P2 , .. • ,Pp'}'

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a d c' ,e" is an isomorphism since rar = 1 for a En. We have clearly for h, hi E Jie' _ { dc' ,e;,(hh' ), if e" = e~ , dc' e,,(h)de, e,,(h l ) - • " I , l' 1 0, If e I- el . Hence, the map which associates to any square matrix (he' ,e") with entries in - , II h ~ eHe' indexed by (e ,e )E.9x.9,theelement E(e',e")E.9'X.9'de',e"( e',e,,)EH defines a ring isomorphism (a) (Jie)n ~ H (see 8.7(e)). Here

He[r(e)] = €a ~Tw ,eY' wEwt yEne)

We define an isomorphism He[r(e)] ~ eHe by sending fTw,eY (f E~) to f~r;. From Lemma 8.13(c) and (d) and Lemma 8.14, we see that this is compatible with the ring structures. Combining this with (a), we get a ring isomorphism Hclr( e)]n ~ H. From the definitions, it follows easily that this is an isomorphism of tiwo -algebras.

8.17. Remark. The isomorphism we have constructed depends on the choice of sequences ,(e'). However, another choice will only change the isomorphism into its composition with an inner automorphism of H(e)[r(e)]n defined by an n x n diagonal invertible matrix.

9. SECOND REDUCTION THEOREM 9.1. We preserve the setup of §3. Assume that we are given a Wo-orbit 1: in :T and an element Vo E C· of infinite order. Let (vo} be as in 8.1. We make the following assumption: (a) if tE1: and a E R, then

(Ja(t) E (Vo} , if ¢ 2Y, { a (Ja(t) E ±(Vo} , if a E 2Y.

Z 9.2. The exponential map e: C - C* (z - e(z) = e ) induces a homomor- phism of complex Lie groups t = Y ® C I~e Y ® C* =:T which will be denoted e. It is Wo-equivariant. We select ro E C such that Vo = e TO • We define a map 1: - t as follows.

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Let t E 1:. For any a E R, let no E Z be such that (Jo(t) = ±v;a (see 9.1(a)). Since Vo is of infinite order, we have

{ no+p = no + n p , whenever a, p , a + PER, no + n_ o = 0, for all a E R.

Hence, there is a unique element l in the C-subspace of t generated by R such that a(l) = noro for all a E R. Then t 1-+ l is the required map 1: --+ t. This map is clearly Wa-equivariant. Hence, its image l: is a Wo-orbit in t. We now define for any tE1: an element to E:T by to = t· e(l) -I . The map 1: --+ :T, t 1-+ to is clearly Wo-equivariant. On the other hand, for each tE1: , (a) to is Wa-invariant (see below). Hence, to is necessarily independent of the choice t (it depends on 1:). Let us prove (a). From the definitions, we have for any a E R,

if a ¢ 2Y, if a E 2Y,

so that (a) follows from Lemma 3.15. We can now define (b) l: --+ 1: (ll-+ toe(l)) , where to is defined as above in terms of any tE1: . Then (c) the map (b) is a Wa-equivariant bijection.

(Its inverse is t 1-+ l.) We shall use the notation (H ,t!J, ... ) and results of §§4 and 5 for the par- ticular to E:T considered above. We shall also use the notation and results of §7 relative to 1: and l: as above. We can state the following result.

~ Theorem 9.3. There are natural isomorphisms of C-algebras % ~ ::t:,- & ~ t!J , !l. ~ H. Moreover, the last two isomorphisms are compatible with the !?- and ::t: -algebra structures, via the first isomorphism.

9.4. For the proof, we shall need a lemma. We fix a En, and we regard ~(a) (resp. g(a)) as ameromorphic function on :TxC" (resp. txC) (see 3.8 and Proposition 4.4). Composing ~ (a) with the holomorphic map "': t x C --+

Z :T x C", (c!, z) 1-+ (toe(c!), e ), we obtain a meromorphic function j(a) on txC.

Lemma 9.S. The meromorphicjunction j(a)g(a)-I is holomorphic and nonva- nishing at all points of l: x {ro} .

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Proof. We have for (e!, z) E t xC,

- 1 (~(a)g(a)- )(e!, Z) ea(~)+2;'(a)z _ 1 a(e!) if a ¢ 2Y, a(e!) + U(a)z • ea(~) - 1 ' = (0 a (to)ea(~)+(;'(a)+;'· (a))z - 1)(0a (to)ea(~)+(;'(a)-;'· (a))z + 1) a(e!) a(e!) + (J.(a) + Oa(tO)J.*(a))z e2a(~) - 1 ' if a E 2Y. Recall that 0a(tO) = ±1 if a E 2Y . Hence, in that case we can also write (j(a)g(a)-I)(e!, z) = ea(~)+(;'(a)+8Q(to);'·(a)) - 1 . ea(~)+(;'(a)-8Q(to);'·(a))z + 1 a(e!) a(e!) + (J.(a) + Oa(tO)J.*(a))z ( ) e2a(~) - 1 .

To show holomorphicity, it is enough to show that for all t E t (a) 2niZ- {O}, if ¢ 2Y, { a a(t) ¢ nloz -{O} , 1Of· a E 2Y ° To show nonvanishing, it is enough to show that for all t E t (b) a(l) + 2J.(a)ro ¢ 2niZ - {O}, if a ¢ 2Y , a(l) + (J.(a) + Oa(tO)J.*(a))ro ¢ 2niZ - {O}} if a E 2Y. a(l) + (J.(a) - Oa(tO)J.*(a))ro ¢ 2ni(Z+!) , Substituting a(l) = nro (n E Z), we see that if one of the statements (a) or (b) IS. VI0'·1 ate, d we would h ave ro = 2 nln° '/ n /I lor~ some nonzero .,mtegers n , n /I . Hence, vo = ero would be a root of 1, contradicting our assumptions in 9.1. 9.6. Proof of Theorem 9.3. Since 'II in 9.4 is locally a holomorphic isomor- phism, it defines for each t E t an isomorphism

...... an e.... .:::::...an ~t5-, t5 I,vo I,ro where t = toe(l). Taking the direct sum over all t E t and using 9.2(c) we get _ -an an isomorphism EBIEl:t5~~o --+ EB fEl:t5I ,ro' Using 7.6(a) and 7.5(a) and (b), this can be regarded as an isomorphism & ~ ;;. This is clearly compatible with the Wo-actions. Taking Wo-invariants, we get an isomorphism !? ~ %. ~ ~ ~ We also get an isomorphism j: K ~ K of the full rings of quotients of t5, t5, which is again Wo-equivariant. We define /: HF --+ HF by

(kw E K) (see 7.4(a)).

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This is compatible with multiplication:

j'(kw Tw • kw' Tw') = j' (kw w(kw' )Tw Tw') = j'(kw w(kw' )Tww') = j(kww(kw,))fww' = j(kw)wU(kw,))fw fw' = j(kw)fwi(kw,)fw' = j' (kw Tw)j' (kw,Tw')' (We have used Proposition 5.2.) ~ ~

We will show that j' maps H (c HF ) isomorphicallyonto H (c HE)' From 7.4(b), we know that H is the subring of HF generated by ~a + 1 ~ ~ ~ (a E TI) and by ~ and that H is the subring of H F generated by tSa + 1 (a E TI) and by ;;. Moreover, by definition, j' defines an isomorphism Iff -?;; and j'(~a + 1) = j'((TO + 1)~(a)) = (fo + l)j(~(a)) = (tSa + l)g(a)-1 j(~(a)).

~ence, it is enough to show that j(~(a))g(a)-l E K is an invertible element of ~, for any a En. But this follows from Lemma 9.5. The theorem is proved. 9.7. In the previous results we have assumed (see 9.1) that Vo is of infinite order. Analogous results hold for Vo = 1. In this case, if 1: is a »a-orbit in !T satisfying 9.1(a), then by Lemma 3.15, 1: consists of a single (»a-invariant) element to' We define '0 = 0 and 1: = {O} ct. The bijection 9.2(b) continues to hold. The proof of Lemma 9.5 applies without change (the left-hand sides of9.5(a), (b) are all zero). Hence, the statement (and proof) of Theorem 9.3 continues to hold (without change).

10. ON SIMPLE H -MODULES 10.1. For any ring A, we denote by Irr A the set of isomorphism classes of simple A-modules. If M is an A-module, then M EB··· EB M (n copies) can be regarded in an obvious way as an An-module (An as in 8.5), and M -? M EB ... EB M defines a bijection (a) IrrA~IrrAn' 10.2. We preserve the setup of §3. If M is a simple H-module, then % acts on M by scalars (by a well-known version of Schur'S lemma due to Dixmier). Hence, there is a unique maximal ideal Jcr.,vo) of % (see 7.2) such that JC1:.,vo)M = O. This defines a partition (a) Irr H = l1(l:,vo) Irr(l:,Vo)H, where 1: runs over all »a-orbits in !T and Vo runs over C· ; Irr(l:,Vo)H consists of those M for which J(l:,vo)M = O.

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10.3. We now fix (1:,vo) as above, and let %, H be the corresponding completions of % , H (see 7.3). We denote by ~I;,Vo) the unique maximal ideal of %.- For any %- -algebra A, we denote by Irro A the set of all M I E Irr A for which ~I;,Vo)M' = O. We have

H/J(I;,vo)H = H/~I;'Vo)H (a finite-dimensional algebra over C). This defines a bijection (a) Irr(I;,vo)H ~ Irro H .

10.4. We now assume that Vo is of infinite order in C". We select '0 E C such that e'o = vo' We partition 1: in equivalence classes (in terms of vo) as in 8.1, and we select an equivalence class c. By Theorem 8.6, we have an isomorphism of %-algebras (a) H ~ Hclr(c)]n (n = #9', as in 8.1). By definition of c, the hypothesis of §9 (see 9.1(a)) is satisfied if H, Wa, 1:, Vo are replaced by He' W~, c, vo' Hence, the constructions of §9 are applicable. In particular, the construction in 9.2 applied to c instead of 1: provides us with an element to E g-wt and a W~ orbit c C t (instead of f) such that e: c ~ c. We define the graded algebra He associated to He and ~ ~ this to as in §4. Let %e- (resp. He or &c) be the completions of the center %e of He (resp. of He itself or &) with respect to the maximal ideal J of %e determined by (c"o) (as in 7.3). Similarly, let ~, He be the completions of ~, He with respect to the maximal ideal of ~ determined by (c, vo)' By Theorem 9.3, we may identify naturally ~ = %c' and we have a natural isomorphism (b) He ~ He of ~- (or %c-) algebras. Now r(c) acts on He' and this induces an action of r(c) on He' Moreover, from the definition of r(c) and of to' we see that to is r(c)-invariant. Hence, the r(c)-acti~n on He induces a r(c)-action on the associated graded algebra He and on He' Now (b) is compatible with the r(c)-actions (using the definitions). Hence, it extends to an isomorphism (c) He[r(c)] ~ He[r(c)] taking 1· Y to 1· Y for y E r(c). Then (c) is an isomorphism of algebras over -r(e) -r() - %c = % e e = % (see 8.5). To simplify notation, we denote by H' the C-algebra Hc[r(c)]. Let %' be the center of H' .

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 634 GEORGE LUSZTIG ,-Wo(e) . Lemma 10.5. We h ave % = & (Wa(C) as In 8.1). This is a result of the same type as Proposition 4.5. Its proof follows almost word for word the proof of [5, 6.5]. 10.6. From 10.5, we see that the maximal ideals of %' are in 1-1 correspon- dence with the Wo(c)-orbits in txC. Now (c, ro) is such a Wa(c)-orbit. Hence, it defines a maximal ideal l' of %' . ;:;::;, .::::..' ~, " Let % , & , it denote the J -adic completions of the % -algebras :z , &, H', and let J' be the unique maximal ideal of !i' . .- - _I.- We have as in 7.5(b), &c ~ EaiEC &' and similarly, & = EalEC&l. Hence, ~, ~ (a)&=&c· Taking Wa(c)-invariants in (a), we obtain (b) !?' = (%c)ro(e) . As we have seen in 10.4, we have (% clo(e) = (~l(e) =!? Hence, from (b) we deduce (c) %-, =%.- . -;-, ~ In partIcular, J = J(r.,vo) • Now (a) shows that the 7-adic and l'-adic completions of & coincide. Since Hc is a free left &-module, it follows that the 7-adic and 1'-adic completions of H c coincide. It also follows that -, ..:::::::.. (d) H = Hc[r(c)]. Let Irr(C ,TO) H' be the set of all M' E Irr H' such that J'M' = O. We have

H' / l'H' = it/ ]'H' (a finite-dimensional algebra over C). Hence, we have a natural bijection (e) Irr(c,ro)H' ~ IrroH'. 10.7. From 10.4(c), 10.6(d) and (e), we obtain a natural bijection IrroHJr(c)] ~ Irr(c,ro) H'. Using 10.I(a), this gives rise to a bijection Irro He[r(c)]n ~ Irr(c,ro) H' . Combining this with 10.4, we obtain a bijection ~ , Irro H ~ Irr(C ,ro) H . Using this and 10.3(a), we obtain the main result of this paper. Corollary 10.S. Recall that Vo is assumed to have infinite order. There is a natural bijection

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10.9. The same proof provides (for Vo of infinite order) an equivalence of categories between ModCE,vo) H (the category of H-modules of finite dimension over C, annihilated by some power of '('I.,vo)) and Mod(c,ro)Hc[r(c)] (the category of Hc[r(c)]-modules of finite dimension over C, annihilated by some power of 1'). The dimension of the module in the first category is # dim.9 times the dimension of the corresponding module in the second category. This remains true when Vo = 1 (we then take '0 = 0), see 9.7. In this case, we have c = {O}, r(c) = {I}.

REFERENCES

1. D. Kazhdan and G. Lusztig, Pro%/ the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153-215. 2. R. Kilmoyer, Principal series representations o/finite Chevalley groups, J. Algebra 51 (1978), 300-319. 3. G. Lusztig, Singularities. character formulas and a q-analog o/weight multiplicities, Asterisque 101-102 (1983), 208-229. 4. _, Some examples o/square integrable representations o/semisimple p-adic groups, Trans. Amer. Math. Soc. 277 (1983), 623-653. 5. _, Cuspidallocal systems and graded Hecke algebras I, Inst. Hautes Etudes Sci. Pub!. Math. 67 (1988), 145-202. 6. I. G. Macdonald, Spherical functions on a group 0/ p-adic type, Pub!. Ramanujan Inst., no. 2, Madras, 1971.

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